Problem: Rewrite as a simplified fraction. ${1.91\overline{7} = {?}}$
Answer: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}1000x &= 1917.7777...\\ 100x &= 191.7777...\end{align*}}$ Subtracting the two equations, we have: ${900x = 1726}$ ${x = \dfrac{1726}{900}} $ ${x = \dfrac{863}{450}} \text{or} {1 \dfrac{413}{450}}$